Explicit construction of Gevrey Quasi-Periodic Discrete Schrödinger operators with cantor spectrum

We construct 1-dim difference Schrödinger operators with a class of Gevrey potentials such that the Cantor spectrum occurs together with the estimations of open spectral gaps for $ 0<|\lambda|\leq 1 $. The proof is based on KAM1 and the Moser-Pöschel argument.1KAM is the abbreviation of the Kolmogorov-Arnold-Moser Theorem.